Differential Calculus

1. List and apply properties of real numbers.
2. Define and identify different types of functions.
3. Determine the existence and value of limits.
4. Define and identify continuous functions.
5. Calculate derivatives of various functions.
6. Apply differentiation techniques to solve real-world problems.

13 weeks, about 60 hours in total. Yours will differ.

1. Week 1Module 1:

   • Unit 1: Basic Properties of Real Number · 3 hours

     Review the definition of sets and their notations.. Study the different types of real numbers: natural, integers, rational, irrational, and real numbers.. Understand the basic axioms of real numbers: field axioms and order axioms.. Solve problems involving intervals and absolute values..

   • Unit 2: Basic Properties of Real Numbers · 3 hours

     Solve inequalities and represent solutions using intervals.. Practice problems involving absolute values and their properties.. Work through tutor-marked assignments to reinforce understanding..

2. Week 2Module 1:

   • Unit 3: Characteristics of Functions · 4 hours

     Define a function and identify its domain and range.. Study the different types of functions: constant, polynomial, algebraic, and transcendental.. Learn about exponential and logarithmic functions and their properties.. Sketch graphs of basic elementary functions..

3. Week 3Module 1:

   • Unit 4: Limits · 4 hours

     Investigate the characteristics of functions: even, odd, periodic, monotonic, and bounded.. Define and identify inverse functions.. Define and construct composite functions.. Determine whether a function has an inverse..

4. Week 4Module 1:

   • Unit 5: Algebra of Limits · 5 hours

     Study the formal definition of a limit of a function.. Prove that the limit of a function is unique.. Evaluate the limit of a function using various techniques.. Evaluate right-hand and left-hand limits.. Use the epsilon-delta method to prove that a number is the limit of a function..

5. Week 5Module 2:

   • Unit 1: Algebra of Limits · 5 hours

     State and apply theorems on limits: sum, product, and quotient theorems.. Evaluate limits of functions using the sum, product, and quotient theorems.. Evaluate limits of functions as x approaches infinity and negative infinity.. Work through examples involving algebraic manipulation to find limits..

6. Week 6Module 2:

   • Unit 2: Differentiation · 4 hours

     Define a continuous function at a point.. Recall properties of continuous functions.. State theorems on continuous functions.. State the three conditions for continuity of a function at a given point.. Identify points of continuity and discontinuity of a function..

7. Week 7Module 2:

   • Unit 3: Rules for Differentiation I · 5 hours

     Define the slope of a point on a curve.. Define the derivative of a function at a given point.. Evaluate the derivative of a function using the limiting process (delta process or from first principles).. Derive standard formulas for differentiation of polynomials.. Find the derivative of polynomial functions using the delta process or a standard formula..

8. Week 8Module 2:

   • Unit 4: Rules for Differentiation II · 5 hours

     Derive the following rules for differentiation: sum rule, difference rule, product rule.. Differentiate all types of polynomial functions using these rules.. Practice applying the product rule and quotient rule to various functions.. Solve problems involving combinations of differentiation rules..

9. Week 9Module 3:

   • Unit 1: Further Differentiation · 5 hours

     Apply the chain rule to differentiate composite functions.. Differentiate logarithmic functions.. Carry out logarithmic differentiation.. Differentiate exponential functions.. Find the derivative of the function a^u..

10. Week 10Module 3:

    • Unit 2: Differentiation of Logarithmic Functions and Exponential Function · 5 hours

      Differentiate trigonometric functions: sin x, cos x, tan x, etc.. Differentiate inverse trigonometric functions: arcsin x, arccos x, arctan x, etc.. Differentiate hyperbolic functions: sinh x, cosh x, tanh x, etc.. Differentiate inverse hyperbolic functions..

11. Week 11Module 3:

    • Unit 3: Differentiation of Trigonometric 41 Functions · 4 hours

      Use the first derivative to determine where a curve is increasing, decreasing, or stationary.. Use the second derivative to determine where a curve is concave upwards or concave downwards.. Identify points of inflection.. Sketch curves using information from the first and second derivatives..

12. Week 12Module 3:

    • Unit 4: Differentiation Inverse Trigonometric Functions and Hyperbolic Functions · 4 hours

      Define global and local minimum and maximum values.. Apply differentiation to solve maximum and minimum problems.. Solve rate problems using differentiation.. Work through examples involving optimization and related rates..

13. Week 13Module 4:

    • Unit 1: Curve Sketching · 4 hours

      Use differentials to approximate values of functions.. Apply differentiation to calculate velocity and acceleration of moving bodies.. Solve problems involving approximations, velocity, and acceleration.. Review all tutor marked assignments..